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\title{\huge 
	Representing Homology Classes by Locally Flat	Surfaces of Minimum Genus%
\thanks{This is an excerpt from a paper published under the same title
in the American Journal of Mathematics \textbf{119} (1997), 1119--1137.
Typeset by the authors using \LaTeX\ 
with packages from \AmS\ and \Xy-pic.}
}
\author{Ronnie Lee and Dariusz M.~Wilczy\'{n}ski\\
\small\scshape Yale University\\
\small\scshape Utah State University}
\date{}
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\section{Introduction}

A necessary and sufficient condition will be given for a nontrivial homology classof a simply connected \text{4-manifold} to be represented by a simple, topologicallylocally flat embedding of a compact Riemann surface.

\section{Splittings of Hermitian Modules}

We begin with an algebraic result.

\begin{thm}
The following is a commutative diagram of pointed hermitian modules.\\ %\CompileMatrices \[\xymatrix{ (M,h,z) \ar[dd]^{\pi_0} \ar[dr]^\alpha_\cong \ar[rr]^{\pi_1} && (M_1,h_1,0) \ar'[d]^-{\pi_{1d}}[dd] \ar[dr]^{\alpha_1}_\cong \\ & (M',h',z')\oplus H(\Lambda^k) \ar[dd]^<(.25){\pi_0} \ar[rr]^<(.25){\pi_1} && (M'_1,h'_1,0)\oplus H(\Lambda_1^k) \ar[dd]^{\pi_{1d}} \\ (M_0,h_0,z_0) \ar@{=}[dd] \ar[dr]^{\alpha_0}_\cong \ar'[r]^<(.6){\pi_{0d}}[rr] && (M_d,h_d,0) \ar@{=}'[d][dd] \ar[dr]^{\alpha_d}_\cong \\ & (M'_0,h'_0,z'_0)\oplus H(\Lambda_0^k) \ar[dd]^<(.25){\beta'_0\oplus\text{id}}_<(.25)\cong\ar[rr]^<(.25){\pi_{0d}} && (M'_d,h'_d,0)\oplus H(\Lambda_d^k) \ar[dd]^{\beta'_d\oplus\text{id}}_\cong \\ (M_0,h_0,z_0) \ar[dr]^{\beta_0}_\cong \ar'[r]^<(.6){\pi_{0d}}[rr] && (M_d,h_d,0) \ar[dr]^{\beta_d}_\cong \\ & (L,\lambda,x)\oplus H(\Lambda_0^k) \ar[rr]^{\pi_{0d}} && (L_d,\lambda_d,0)\oplus H(\Lambda_d^k) }\]
\end{thm}

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